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Here's something similar to what you conjecture. Let's work instead with functions $\mathbb{R} / 2\pi \mathbb{Z} \to \mathbb{C}$; one can do similar things in the real-valued case but I find the comp …

It's not precisely what you asked about, but this paper by Gluskin and Milman shows that, for "most" sequences $a_1, \dotsc, a_n$, the AM-GM inequality can be reversed up to a multiplicative constant. …

A classical result along these lines is Bonnesen's inequality, which states $$ L^2 - 4\pi A \ge \pi^2 (r_{out} - r_{in})^2, $$ where $L$ is the length and $A$ is the enclosed area of a simple planar c …

This is a long comment rather than an answer. In this paper of Klartag and Milman, the following operation on functions is defined and called the Asplund sum (as far as I can tell, that name has not …

"Everyone" knows that for a general $f\in L^2[0,1]$, the Fourier series of $f$ converges to $f$ in the $L^2$ norm but not necessarily in most other senses one might be interested in; but if $f$ is rea …

In the context of 3), what I have heard from folklore is that when (1) holds, the Kolmogorov distance (not total variation) is bounded by something like $1/\sqrt{m}$. This bound follows if (1) holds …

Regarding the last part of the question, I haven't looked at either of the following books myself, but I've seen them referred to for systematic presentations of the theory of quasinormed spaces (whic …